Origami & Folding
what one sheet can do
Concept

Crease radius — where it appears

The width of the rounded region a real crease occupies, rather than the line an idealised one is. It costs paper at every fold, and the loss accumulates across a grid in a way that limits how fine a pattern can be.

Named by 18 essays across 5 fields — each of them below, with the objects they name alongside it.

no thicknesslayers add up; a 64-grid model is millimetres thick at the coreno stretchpaper stretches a little, which is why wet-folding works at allcreases are linesa crease has a radius; sharp folds tear and soft ones springperfect memorypaper relaxes, so a model opens slightly the moment it is put downthe theorems are exact statements about a sheet nobody has ever folded

Four things that are not true

Zero thickness, no stretch, creases that are lines, and perfect memory. Every theorem on this site rests on all four, every one of them is false, and the interesting engineering is exactly where each fails.

material · Idealisation ρ = 0.12 mmarc 0.377 mm, stack advances 0.240 mmlost per crease (π − 2)ρ = 0.1370 mmone crease, at its real radiuson a 150 mm sheet8 × 81.0 mm — 0.6%16 × 162.1 mm — 1.4%24 × 243.2 mm — 2.1%32 × 324.2 mm — 2.8%48 × 486.4 mm — 4.3%lost along every line of the grid, in both directionswhich is why an ambitious grid is folded from thin paper

The crease has a radius

A fold does not go through a line. It goes round a small arc, and the arc uses more paper than the stack advances by — a fraction of a millimetre per crease, and several millimetres across a grid, which is why an ambitious tessellation comes out short.

material · Idealisation 24681012-202halvingsmetres of paper (powers of ten)297 mm — 6 folds, 64 layers1 m — 7 folds, 128 layers10 m — 8 folds, 256 layers100 m — 10 folds, 1024 layers1200 m — 12 folds, 4096 layerspaper 0.1 mm thick · L = (πt/6)(2ⁿ + 4)(2ⁿ − 1)the loss is the paper that goes round the closed end, and it doubles twice per fold

How many times can it be halved

The folklore says seven, and the folklore is a statement about one sheet of paper. What actually binds is arithmetic: every halving doubles the layers and the paper spent at the closed end grows as the square of the layer count, so the length needed for twelve folds is nearly a kilometre.

material · Idealisation 010203040506000.20.40.60.81foldsshare of the sheet lost to hinges50% of the sheet64 foldshinge radius 0.08 on a 10 unit sheet · (π − 2)ρ = 0.0913 lost per fold

Nothing in a body folds on a line

A crease in an organism is not a crease. It is a compliant region — a patch of thinner material that bends — and a region has a width. The width consumes surface in exact proportion to the number of folds, which puts a ceiling on how fine a pattern can usefully get.

biology · Insect wings what each round of folding reaches, and how close together it ison a sheet 150 mm squaremarksclosest pairmedian gapwithin 0.2 mmone fold, every axiom975.000 mm75.000 mm0.0%two folds, every axiom5650.520 mm2.700 mm0.0%three folds, point onto point only5538230.000 mm0.058 mm94.4%a crease in ordinary paper is about that wide, so the last column is the share of marks a folder cannot separate

Closer than a crease is wide

One fold from a bare square leaves nine marks, seventy-five millimetres apart. Two folds leave five hundred and sixty-five, the closest pair half a millimetre apart. Three folds — using one axiom of the seven — leave half a million, and ninety-four per cent of them have another mark within a fifth of a millimetre. What bounds a folder is not what the axioms reach; it is what the paper can tell apart.

construction · Reference points the bar is the vertices the drawing has and the list does notThe preliminary base09 listed · panels closeThe Miura fold035 listed · panels closeThe square twist016 listed · panels closeThe hexagon twist022 listed · panels closeThe Yoshimura pattern045 listed · panels closeFold and cut — the triangle011 listed · panels closeThe tapered corrugation040 listed · panels closeThe waterbomb tessellation041 listed · panels closethe square grid, assembled064 listed · panels closethe triangular grid, assembled1282 listed · panels 1.73 apartthe honeycomb, assembled1884 listed · panels 2.00 apartthe rhombille tiling, assembled12138 listed · panels 1.86 apartthe elongated triangular tiling, assembled576 listed · panels 1.73 apartevery pattern with a bar has panels that cannot be placed, and every pattern without one places exactly

How deep is a crossing

A crossing is a verdict with no middle: two creases either pass through one another or they do not, and the first makes a pattern unfoldable while the second leaves it untouched. Measured on the patches where they occur, the shallowest crossing runs 0.16 mm past the end of the crease it meets, on a sheet 150 mm across. Five of the forty-seven are under half a millimetre, which is thinner than the line a pencil draws.

rigid · Tolerance 050100150200250300350020406080100120foldspacking ratio deliveredρ 0.02 — 219 foldsρ 0.05 — 88 foldsρ 0.1 — 44 foldsρ 0.2 — 22 foldssheet of side 10 · D(k) = k(1 − k(π−2)ρ⁄S) · k* = S ⁄ 2(π−2)ρ, and the ceiling it reaches is k*⁄2

Four materials, four optima

The convergence argument gets its pattern and stops there. A hinge has a radius, the radius takes a fixed length of surface out of every fold, and the fold count that gets the most packing out of a sheet is inversely proportional to it — so a leaf, a wing, a gut lining and a metal array agreeing on a corrugation still disagree by an order of magnitude about how many creases to put in one.

biology · Convergence 0102030405060010203040fold half-angle from shut (degrees)packing ratioa hinge stops here — 6°corrugation, closes in one directioncapped at 10×Miura, closes in two at oncecapped at 92×a quoted 8× is 7.2° corrugation or 20.7° Miuraa quoted 15× is 3.8° corrugation or 15.0° Miuraa quoted 30× is 1.9° corrugation or 10.5° Miuraratio = 1 ⁄ sinθ for a corrugation and 1 ⁄ sin²θ for a Miura · θ is the half-angle from shut · a hinge that stops at 6° is the dashed line

The number is the angle

Every packing ratio worked out so far is computed at a fold closed all the way, and a folded wing is not closed all the way. At zero thickness the ratio runs away as the fold shuts, so the size of a quoted number says how far the fold got and not what the pattern is — and the pattern contributes only an exponent, which makes the same quoted ratio mean two quite different angles depending on which geometry produced it.

biology · Insect wings 02040608000.20.40.60.81half-angle from shut (degrees)share of the sheet's lengthspan, every fold countclearance, 4 foldsclearance, 8 foldsclearance, 16 foldssheet 10 · the span is S·sin θ at every fold count; the clearance is (S⁄k)·cos θ and falls as the count rises

The fold count sets the spring

A corrugation sweeps the same span at every fold count, and the count decides only how much room the zigzag needs while it does it — which was counted as a gain with nothing pushing back. Something does push back. Every hinge is a spring, a finer corrugation has proportionally more of them, and the force to hold a given span rises exactly as the clearance falls: the product of the two is the same number whatever the count, and at each material's own best count the spring goes as one over the square of the hinge radius.

biology · Convergence 00.20.40.60.81050100150span held, as a share of the sheetforce needed to hold it there4 folds8 folds16 foldsthe pale line under each: what it takes at flat, 4kc ⁄ Ssheet 10, hinge radius 0.05 · the corrugation springs open, and this is what has to stop it at each state

How far open is a question about the grip

A corrugation of hinges that rest flat is loaded when it is shut, so it opens by itself and the force in the held-state calculation is a restraint rather than a drive. Followed from shut to flat that restraint only ever falls, and by exactly π over two — so every partly open state a structure can occupy is squeezed into a band a third wide, and a grip that weakens by a third leaves the sheet nine tenths open.

biology · Convergence -3-2-1123050100150turn in each hinge (radians, signed by branch)energy storedrests hereand hereflat: 179the barrier isthe flat state8 folds, hinge 0.05creased to 1.6the two branches are the same sheet folded opposite ways, and the flat sheet is the only state they share

Springs that disagree do not offer a choice

A corrugation of hinges that remember different angles was expected to have more than one position in which nothing pushes. It has exactly one, at the stiffness-weighted mean of what they remember, because a sum of parabolas in one variable is a parabola. The second resting place comes from somewhere else entirely — the mirror pattern — and the flat sheet is the barrier between them, which is also why a creased sheet cannot be pulled flat at all.

biology · Convergence folding, priced by how often it has to happena crease of radius ρ strains its outer fibre by t ⁄ 2ρ, and a material takes less strain the more often it is askedcycles it must survivesmallest hinge radiusbest fold countpacking it reachesagainst once10.05087.643.8—100.15827.713.93× worse1000.5008.7604.38010× worse1,0001.5812.7701.38532× worsesheet 10, thickness 0.1, fatigue exponent 0.5 · the radius goes as N^0.5 and the packing as N^−0.5

What a second deployment costs

Every folded structure this field builds deploys once. The reason is a power law: a hinge asked to survive more cycles cannot be as sharp, a blunter hinge takes more surface out of the sheet, and the fold count that packs best falls as the cycle count to a fatigue exponent. A structure required to work a thousand times packs thirty times worse than one required to work once, and the exponent decides how fast rather than whether.

rigid · Deployables what a finer pattern costs in confidenceeach hinge working 0.999 of the time, against a target of 99 per cent for the whole deploymenthingesthe system openseach hinge needssuccesses to show itand the article itself899.2%0.9987442,385cannot be tested2497.6%0.9995817,154cannot be tested6094.2%0.99983317,885cannot be tested12088.7%0.99991635,769cannot be tested30074.1%0.99996689,422cannot be testedr consecutive successes put a 95 per cent lower bound of 0.05^(1⁄r) on a hinge, and a flight article deploys once

The crease count is a reliability budget

A deployment that needs every hinge to work is the hinge reliability raised to the crease count, so the fineness that buys compaction spends the probability of getting it. At a thousandth of a chance of a hinge failing, sixty hinges give a 94 per cent deployment and three hundred give 74. The fold count that maximises expected compaction is well below the one that maximises compaction — and demonstrating the result takes tens of thousands of successful tests on an article that itself deploys once.

rigid · Deployables how many fibres thick each paper is25 microns to a fibrethe fibre width is stated rather than measured here, and the ordering survives any figure near itcopier paper4.0100 µm · 80 g/m² · 30 layers of stackkami2.870 µm · 60 g/m² · 42 layers of stacknewsprint2.665 µm · 45 g/m² · 46 layers of stackwashi1.640 µm · 30 g/m² · 75 layers of stackfoil-backed tissue1.026 µm · 22 g/m² · 115 layers of stackunryu tissue0.718 µm · 12 g/m² · 166 layers of stacka sheet one fibre thick has nothing through its thickness to hinge, which is why the thinnest here are backed rather than folded

The paper that will not hold a crease

Every constraint these essays have found improves as the paper gets thinner: the stack, the size, the layer count. A crease does not. A crease is a plastic hinge in the fibres at the fold, and a sheet one fibre thick has nothing through its thickness to hinge — so there is a floor under the thickness that no manufacturing skill moves, because the fibre diameter is a constant of the plant. The papers a tradition folds sit between one and a half fibres and four, and the two below that in this collection's own shelf are tissues, which are backed with foil before anybody creases them.

history · Paper as substrate metres of crease a square metre: what each pattern asks for, and what each paper allowsthe pale bars are patterns and the dark ones are paperswhat foil-backed tissue allows641026 µm, a crease 0.16 mm acrosswhat washi allows416740 µm, a crease 0.24 mm acrosswhat kami allows238170 µm, a crease 0.42 mm acrosswhat copier paper allows1667100 µm, a crease 0.60 mm acrossThe waterbomb tessellation89printed at 160 mmThe Yoshimura pattern82printed at 170 mmThe tapered corrugation41printed at 160 mmThe hexagon twist41printed at 150 mmThe Miura fold36printed at 170 mmThe preliminary base32printed at 150 mmThe square twist31printed at 150 mmFold and cut — the triangle11printed at 150 mma crease occupies about 6 sheet thicknesses, so the closest two creases can be laid is that, and the ceiling is its reciprocal

The density a paper allows

Every density these essays measure is a quotient a pattern hands over, and nothing has asked what the paper's own answer is. It has one: a crease occupies a band a few thicknesses across, so two creases closer than that are not two creases, and a sheet of a given thickness carries a largest density. Copier paper allows 1,667 metres of crease a square metre and the densest pattern on the printed shelf asks for 89 — a factor of nineteen below the worst paper's ceiling. The material is not what limits a crease pattern's density at any fineness anybody folds.

material · Crease density crease density, the closest two creases come without meeting, and the two multipliedthe closest approach is between creases with no vertex in common, measured as segmentspatternm a m²closest mmproductThe Yoshimura pattern8224.52.02The waterbomb tessellation8920.01.79The square twist3136.11.13The Miura fold3625.10.91The hexagon twist4122.10.90The tapered corrugation4118.60.77parallel creases give a product of exactly one; a pattern above one carries more length than its own spacing would suggest

A paper limits spacing, not density

The density ceiling put a sheet's limit at parallel creases a crease-width apart, 1⁄w of crease a square metre, and claimed no arrangement carries more. Crossing families do: they meet at vertices, which is shared ground, and never come closer than w anywhere else. Measured over every pair of creases that do not share a vertex, density times closest spacing settles at about 1.9 for both the Miura and the waterbomb, so each can be folded nearly twice as fine as the density bound said — 262 cells a side on copier paper for the Miura rather than 139, and 141 for the waterbomb rather than 74.

material · Crease density the crease each pattern has least room on, once the ground near its ends is creased twicea narrow sector pushes the overlap out along both creases, as one over the sine of the anglepatternnarrowest sectorreach, both endsthat crease, mmroom to shrinkThe tapered corrugation65.9°2.1020.416×The waterbomb tessellation45.0°2.4128.320×The hexagon twist60.0°2.1525.520×The Yoshimura pattern60.0°2.3128.320×The Miura fold69.9°2.0626.722×The square twist90.0°2.0036.130×Fold and cut — the triangle58.2°1.1828.640×The preliminary base45.0°1.4175.088×reach is in band widths; room to shrink is the crease's length over that reach, on copier paper with a band 0.6 mm wide

A vertex creases the paper twice

Two creases that meet share ground near the point, and their bands overlap out along each of them to w⁄sin θ for a sector angle θ below a right angle. Add the overlap at both ends of a crease, and a crease no longer than that is overlap from end to end — a crease only in the drawing. That third bound binds before the spacing does: the finest Miura on copier paper is 135 cells a side by its vertices against 262 by its spacing, and the finest waterbomb 82 against 141. Both land within a tenth of the density bound, which had the wrong argument and nearly the right number.

material · Crease density a corrugation remembering 0.8 rad, held at 2.4 and later opened to 0.4torques as shares of the one total both ends of the story share; the barrier as a multiple of its creased valueheld for, τremembered turncontainer suppliesopening needsbarrier0.00.80080%20%1.00×0.51.43049%51%3.19×1.01.81129%71%5.13×2.02.18311%89%7.45×3.02.3204%96%8.41×5.02.3891%99%8.92×the container's share and the opening's share always sum to the whole, whatever the holding time

Holding a fold moves the force

A creased hinge held at an angle slowly comes to remember that angle, so a leaf or a wing packed in a bud for a season is gradually holding itself and the bud has less to do. The force is not used up in the process. The torque the container must supply falls as e^(−T⁄τ), the torque later needed to open the structure rises by exactly the same amount, and the two sum to the same total at every moment of the holding. Held for three relaxation times, a corrugation creased to 0.8 radians and packed to 2.4 needs 4 per cent of the total from its container and 96 per cent from whatever opens it — and the barrier to its mirror image has grown more than eightfold. A packing that lasts buys independence from its container with a harder unfolding.

biology · Convergence

Named alongside it

The objects these essays reach for when they reach for this one.

IdealisationCorrugationThicknessTrade-offConvergenceMembrane hingePacking ratioDeploymentError propagationCrease lengthEnergy minimisationInsect wings

All concepts